I am currently working at a math camp filled with tiny math geniuses, and one of them gave me a problem yesterday with which I am having trouble. This problem was written by a fourth grader and has reportedly only been solved on one other occasion. I'm not particularly good with patterns, and I wondered if anyone on this site might have any ideas.
"On Tuesday, it was very rainy, as Jane noticed. In the first minute, there were only 2 drops, while there were 7 drops in the second minute. There were 33 drops in the third minute, but that soon escalated into 266 drops in the fourth minute and eventually 3140 drops in the fifth minute. If this pattern continues, it will be very rainy. Help Jane figure out how many drops fell total in the first ten minutes. Based off your answer, tell Jane whether to play outside or not."
The key part to this, in my opinion, is that "if the pattern continues, it will continue to be rainy." However, that doesn't mean that the inside beginning part of the pattern is definitely always increasing. Yes, it looks that way from the beginning, but what if it's actually some polynomial that has some twists and turns in the beginning, but then skyrockets after $t=5 \text{ min}$?
We have five points, so we can fit them exactly into a quartic. It kind of loops back and forth between $t=1$ and $t=4$, but then it just skyrockets out. As it turns out, for $t=6,7,8,9,10$, this polynomial yields integer values for $y$ despite the fractional coefficients, so we have: $$2+7+33+266+3140+13337+37787+85668+168406+299675=608321\text{ drops}$$ Now, you'll notice that the ratios start to mellow out because this is quartic, not exponential, but it is still very rainy, so I think that this is a suitable answer given the problem description.